Question

Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\frac{p{x}^2+qx+r}{ax+b}$

Answer 1

Let $f\left(x\right)=\frac{p{x}^2+qx+r}{ax+b}$
Thus by quotient rule,
$f^{\prime }\left(x\right)=\frac{(ax+b)\frac{d}{dx}(p{x}^2+qx+r)-(p{x}^2+qx+r)\frac{d}{dx}(ax+b)}{{(ax+b)}^2}$
$=\frac{(ax+b)(2px+q)-(p{x}^2+qx+r)\left(a\right)}{{(ax+b)}^2}$
$=\frac{2ap{x}^2+aqx+2bpx+bq-ap{x}^2-aqx-ar}{{(ax+b)}^2}$
$=\frac{ap{x}^2+2bpx+bq-ar}{{(ax+b)}^2}$